Desigualdad (matemáticas)

Este artículo trata sobre las relaciones "mayor que" y "menor que". Para la relación "no igual", consulte Designación .

"Menor que" y "Mayor que" redireccionan aquí. Para el símbolo <, vea el signo menor que . Para el símbolo>, consulte el signo mayor que . Para la canción de Nine Inch Nails, vea Less Than (canción) .

Mathematical relation expressed by symbols < or ≤

Este artículo incluye una lista de referencias generales , pero permanece en gran parte sin verificar porque carece de suficientes citas en línea correspondientes . Ayude a mejorar este artículo introduciendo citas más precisas. ( Mayo de 2017 ) ( Obtenga información sobre cómo y cuándo eliminar este mensaje de plantilla )

Las regiones factibles de programación lineal están definidas por un conjunto de desigualdades.

En matemáticas , una desigualdad es una relación que hace una comparación no igual entre dos números u otras expresiones matemáticas. ^{[1] [2]} Se utiliza con mayor frecuencia para comparar dos números en la recta numérica por su tamaño. Hay varias notaciones diferentes que se utilizan para representar diferentes tipos de desigualdades:

• La notación a < b significa que a es menor que b .
• La notación a > b significa que a es mayor que b .

En cualquier caso, a no es igual a b . Estas relaciones se conocen como desigualdades estrictas , ^[2] lo que significa que a es estrictamente menor o estrictamente mayor que b . Se excluye la equivalencia.

A diferencia de las desigualdades estrictas, existen dos tipos de relaciones de desigualdad que no son estrictas:

• La notación un ≤ b o un ⩽ b medios que una es menor que o igual a b (o, equivalentemente, a lo sumo b , o no mayor que b ).
• La notación un ≥ b o un ⩾ b medios que una es mayor que o igual a b (o, de manera equivalente, al menos, b , o no menos de b ).

La relación "no mayor que" también se puede representar mediante a ≯ b , el símbolo de "mayor que" dividido en dos por una barra, "no". Lo mismo es cierto para "no menos de" y a ≮ b .

La notación a ≠ b significa que a no es igual ab , y a veces se considera una forma de desigualdad estricta. ^[3] No dice que uno sea más grande que el otro; que ni siquiera requiere una y b para ser miembro de un conjunto ordenado .

En las ciencias de la ingeniería, el uso menos formal de la notación es afirmar que una cantidad es "mucho mayor" que otra, normalmente en varios órdenes de magnitud . Esto implica que el valor menor puede despreciarse con poco efecto sobre la precisión de una aproximación (como el caso del límite ultrarelativista en física).

• La notación a ≪ b significa que a es mucho menor que b . (En la teoría de la medida , sin embargo, esta notación se usa para la continuidad absoluta , un concepto no relacionado. ^[4] )
• La notación a ≫ b significa que a es mucho mayor que b . ^[5]

En todos los casos anteriores, dos símbolos cualesquiera que se reflejen entre sí son simétricos; a < b y b > a son equivalentes, etc.

Propiedades en la recta numérica

Las desigualdades se rigen por las siguientes propiedades . Todas estas propiedades también son válidas si todas las desigualdades no estrictas (≤ y ≥) se reemplazan por sus correspondientes desigualdades estrictas (<y>) y, en el caso de aplicar una función, las funciones monótonas se limitan a funciones estrictamente monótonas .

Converse

Las relaciones ≤ y ≥ son el uno al otro de conversar , lo que significa que para cualquier números reales una y b :

un ≤ b y b ≥ una son equivalentes.

Transitividad

La propiedad transitiva de la desigualdad establece que para cualquier número real a , b , c : ^[6]

Si a ≤ b y b ≤ c , entonces a ≤ c .

Si alguna de las premisas es una desigualdad estricta, entonces la conclusión es una desigualdad estricta:

Si a ≤ b y b < c , entonces a < c .

Si a < b y b ≤ c , entonces a < c .

Suma y resta

Si x < y , entonces x + a < y + a .

Una constante común c se puede sumar o restar de ambos lados de una desigualdad. ^[3] Entonces, para cualquier número real a , b , c :

If a ≤ b, then a + c ≤ b + c and a − c ≤ b − c.

In other words, the inequality relation is preserved under addition (or subtraction) and the real numbers are an ordered group under addition.

Multiplication and division

If x < y and a > 0, then ax < ay.

If x < y and a < 0, then ax > ay.

The properties that deal with multiplication and division state that for any real numbers, a, b and non-zero c:

If a ≤ b and c > 0, then ac ≤ bc and a/c ≤ b/c.

If a ≤ b and c < 0, then ac ≥ bc and a/c ≥ b/c.

In other words, the inequality relation is preserved under multiplication and division with positive constant, but is reversed when a negative constant is involved. More generally, this applies for an ordered field. For more information, see § Ordered fields.

Additive inverse

The property for the additive inverse states that for any real numbers a and b:

If a ≤ b, then −a ≥ −b.

Multiplicative inverse

If both numbers are positive, then the inequality relation between the multiplicative inverses is opposite of that between the original numbers. More specifically, for any non-zero real numbers a and b that are both positive (or both negative):

If a ≤ b, then 1/a ≥ 1/b.

All of the cases for the signs of a and b can also be written in chained notation, as follows:

If 0 < a ≤ b, then 1/a ≥ 1/b > 0.

If a ≤ b < 0, then 0 > 1/a ≥ 1/b.

If a < 0 < b, then 1/a < 0 < 1/b.

Applying a function to both sides

The graph of y = ln x

Any monotonically increasing function, by its definition,^[7] may be applied to both sides of an inequality without breaking the inequality relation (provided that both expressions are in the domain of that function). However, applying a monotonically decreasing function to both sides of an inequality means the inequality relation would be reversed. The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function.

If the inequality is strict (a < b, a > b) and the function is strictly monotonic, then the inequality remains strict. If only one of these conditions is strict, then the resultant inequality is non-strict. In fact, the rules for additive and multiplicative inverses are both examples of applying a strictly monotonically decreasing function.

A few examples of this rule are:

• Raising both sides of an inequality to a power n > 0 (equiv., −n < 0), when a and b are positive real numbers:

0 ≤ a ≤ b ⇔ 0 ≤ aⁿ ≤ bⁿ.

0 ≤ a ≤ b ⇔ a⁻ⁿ ≥ b⁻ⁿ ≥ 0.

• Taking the natural logarithm on both sides of an inequality, when a and b are positive real numbers:

0 < a ≤ b ⇔ ln(a) ≤ ln(b).

0 < a < b ⇔ ln(a) < ln(b).

(this is true because the natural logarithm is a strictly increasing function.)

Formal definitions and generalizations

A (non-strict) partial order is a binary relation ≤ over a set P which is reflexive, antisymmetric, and transitive.^[8] That is, for all a, b, and c in P, it must satisfy the three following clauses:

1. a ≤ a (reflexivity)
2. if a ≤ b and b ≤ a, then a = b (antisymmetry)
3. if a ≤ b and b ≤ c, then a ≤ c (transitivity)

A set with a partial order is called a partially ordered set.^[9] Those are the very basic axioms that every kind of order has to satisfy. Other axioms that exist for other definitions of orders on a set P include:

1. For every a and b in P, a ≤ b or b ≤ a (total order).
2. For all a and b in P for which a < b, there is a c in P such that a < c < b (dense order).
3. Every non-empty subset of P with an upper bound has a least upper bound (supremum) in P (least-upper-bound property).

Ordered fields

Main article: Ordered field

If (F, +, ×) is a field and ≤ is a total order on F, then (F, +, ×, ≤) is called an ordered field if and only if:

• a ≤ b implies a + c ≤ b + c;
• 0 ≤ a and 0 ≤ b implies 0 ≤ a × b.

Both (Q, +, ×, ≤) and (R, +, ×, ≤) are ordered fields, but ≤ cannot be defined in order to make (C, +, ×, ≤) an ordered field,^[10] because −1 is the square of i and would therefore be positive.

Besides from being an ordered field, R also has the Least-upper-bound property. In fact, R can be defined as the only ordered field with that quality.^[11]

Chained notation

The notation a < b < c stands for "a < b and b < c", from which, by the transitivity property above, it also follows that a < c. By the above laws, one can add or subtract the same number to all three terms, or multiply or divide all three terms by same nonzero number and reverse all inequalities if that number is negative. Hence, for example, a < b + e < c is equivalent to a − e < b < c − e.

This notation can be generalized to any number of terms: for instance, a₁ ≤ a₂ ≤ ... ≤ a_n means that a_i ≤ a_i+1 for i = 1, 2, ..., n − 1. By transitivity, this condition is equivalent to a_i ≤ a_j for any 1 ≤ i ≤ j ≤ n.

When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1/2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1/2.

Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities between adjacent terms. For example, the defining condiction of a zigzag poset is written as a₁ < a₂ > a₃ < a₄ > a₅ < a₆ > ... . Mixed chained notation is used more often with compatible relations, like <, =, ≤. For instance, a < b = c ≤ d means that a < b, b = c, and c ≤ d. This notation exists in a few programming languages such as Python. In contrast, in programming languages that provide an ordering on the type of comparison results, such as C, even homogeneous chains may have a completely different meaning.^[12]

Sharp inequalities

An inequality is said to be sharp, if it cannot be relaxed and still be valid in general. Formally, a universally quantified inequality φ is called sharp if, for every valid universally quantified inequality ψ, if ψ ⇒ φ holds, then ψ ⇔ φ also holds. For instance, the inequality ∀a ∈ ℝ. a² ≥ 0 is sharp, whereas the inequality ∀a ∈ ℝ. a² ≥ −1 is not sharp.^{[citation needed]}

Inequalities between means

See also: Inequality of arithmetic and geometric means

There are many inequalities between means. For example, for any positive numbers a₁, a₂, …, a_n we have H ≤ G ≤ A ≤ Q, where

${\displaystyle H={\frac {n}{{\frac {1}{a_{1}}}+{\frac {1}{a_{2}}}+\cdots +{\frac {1}{a_{n}}}}}}$	(harmonic mean),
${\displaystyle G={\sqrt[{n}]{a_{1}\cdot a_{2}\cdots a_{n}}}}$	(geometric mean),
${\displaystyle A={\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}}$	(arithmetic mean),
${\displaystyle Q={\sqrt {\frac {a_{1}^{2}+a_{2}^{2}+\cdots +a_{n}^{2}}{n}}}}$	(quadratic mean).

Cauchy–Schwarz inequality

See also: Cauchy–Schwarz inequality

The Cauchy–Schwarz inequality states that for all vectors u and v of an inner product space it is true that

${\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle ,}$

where ${\displaystyle \langle \cdot ,\cdot \rangle }$ is the inner product. Examples of inner products include the real and complex dot product; In Euclidean space Rⁿ with the standard inner product, the Cauchy–Schwarz inequality is

${\displaystyle \left(\sum _{i=1}^{n}u_{i}v_{i}\right)^{2}\leq \left(\sum _{i=1}^{n}u_{i}^{2}\right)\left(\sum _{i=1}^{n}v_{i}^{2}\right).}$

Power inequalities

A "power inequality" is an inequality containing terms of the form a^b, where a and b are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises.

Examples

• For any real x,

${\displaystyle e^{x}\geq 1+x.}$

• If x > 0 and p > 0, then

${\displaystyle {\frac {x^{p}-1}{p}}\geq \ln(x)\geq {\frac {1-{\frac {1}{x^{p}}}}{p}}.}$

In the limit of p → 0, the upper and lower bounds converge to ln(x).

• If x > 0, then

${\displaystyle x^{x}\geq \left({\frac {1}{e}}\right)^{\frac {1}{e}}.}$

• If x > 0, then

${\displaystyle x^{x^{x}}\geq x.}$

• If x, y, z > 0, then

${\displaystyle \left(x+y\right)^{z}+\left(x+z\right)^{y}+\left(y+z\right)^{x}>2.}$

• For any real distinct numbers a and b,

${\displaystyle {\frac {e^{b}-e^{a}}{b-a}}>e^{(a+b)/2}.}$

• If x, y > 0 and 0 < p < 1, then

${\displaystyle x^{p}+y^{p}>\left(x+y\right)^{p}.}$

• If x, y, z > 0, then

${\displaystyle x^{x}y^{y}z^{z}\geq \left(xyz\right)^{(x+y+z)/3}.}$

• If a, b > 0, then^[13]

${\displaystyle a^{a}+b^{b}\geq a^{b}+b^{a}.}$

• If a, b > 0, then^[14]

${\displaystyle a^{ea}+b^{eb}\geq a^{eb}+b^{ea}.}$

• If a, b, c > 0, then

${\displaystyle a^{2a}+b^{2b}+c^{2c}\geq a^{2b}+b^{2c}+c^{2a}.}$

• If a, b > 0, then

${\displaystyle a^{b}+b^{a}>1.}$

Well-known inequalities

See also: List of inequalities

Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names:

• Azuma's inequality
• Bernoulli's inequality
• Bell's inequality
• Boole's inequality
• Cauchy–Schwarz inequality
• Chebyshev's inequality
• Chernoff's inequality
• Cramér–Rao inequality
• Hoeffding's inequality
• Hölder's inequality
• Inequality of arithmetic and geometric means
• Jensen's inequality
• Kolmogorov's inequality
• Markov's inequality
• Minkowski inequality
• Nesbitt's inequality
• Pedoe's inequality
• Poincaré inequality
• Samuelson's inequality
• Triangle inequality

Complex numbers and inequalities

The set of complex numbers ℂ with its operations of addition and multiplication is a field, but it is impossible to define any relation ≤ so that (ℂ, +, ×, ≤) becomes an ordered field. To make (ℂ, +, ×, ≤) an ordered field, it would have to satisfy the following two properties:

• if a ≤ b, then a + c ≤ b + c;
• if 0 ≤ a and 0 ≤ b, then 0 ≤ ab.

Because ≤ is a total order, for any number a, either 0 ≤ a or a ≤ 0 (in which case the first property above implies that 0 ≤ −a). In either case 0 ≤ a²; this means that i² > 0 and 1² > 0; so −1 > 0 and 1 > 0, which means (−1 + 1) > 0; contradiction.

However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if a ≤ b, then a + c ≤ b + c"). Sometimes the lexicographical order definition is used:

• a ≤ b, if
  • Re(a) < Re(b), or
  • Re(a) = Re(b) and Im(a) ≤ Im(b)

It can easily be proven that for this definition a ≤ b implies a + c ≤ b + c.

Vector inequalities

Inequality relationships similar to those defined above can also be defined for column vectors. If we let the vectors ${\displaystyle x,y\in \mathbb {R} ^{n}}$ (meaning that ${\displaystyle x=(x_{1},x_{2},\ldots ,x_{n})^{\mathsf {T}}}$ and ${\displaystyle y=(y_{1},y_{2},\ldots ,y_{n})^{\mathsf {T}}}$, where ${\displaystyle x_{i}}$ and ${\displaystyle y_{i}}$ are real numbers for ${\displaystyle i=1,\ldots ,n}$), we can define the following relationships:

• ${\displaystyle x=y}$, if ${\displaystyle x_{i}=y_{i}}$ for ${\displaystyle i=1,\ldots ,n}$.
• ${\displaystyle x<y}$, if ${\displaystyle x_{i}<y_{i}}$ for ${\displaystyle i=1,\ldots ,n}$.
• ${\displaystyle x\leq y}$, if ${\displaystyle x_{i}\leq y_{i}}$ for ${\displaystyle i=1,\ldots ,n}$ and ${\displaystyle x\neq y}$.
• ${\displaystyle x\leqq y}$, if ${\displaystyle x_{i}\leq y_{i}}$ for ${\displaystyle i=1,\ldots ,n}$.

Similarly, we can define relationships for ${\displaystyle x>y}$, ${\displaystyle x\geq y}$, and ${\displaystyle x\geqq y}$. This notation is consistent with that used by Matthias Ehrgott in Multicriteria Optimization (see References).

The trichotomy property (as stated above) is not valid for vector relationships. For example, when ${\displaystyle x=(2,5)^{\mathsf {T}}}$ and ${\displaystyle y=(3,4)^{\mathsf {T}}}$, there exists no valid inequality relationship between these two vectors. Also, a multiplicative inverse would need to be defined on a vector before this property could be considered. However, for the rest of the aforementioned properties, a parallel property for vector inequalities exists.

Systems of inequalities

Systems of linear inequalities can be simplified by Fourier–Motzkin elimination.^[15]

The cylindrical algebraic decomposition is an algorithm that allows testing whether a system of polynomial equations and inequalities has solutions, and, if solutions exist, describing them. The complexity of this algorithm is doubly exponential in the number of variables. It is an active research domain to design algorithms that are more efficient in specific cases.

See also

• Binary relation
• Bracket (mathematics), for the use of similar ‹ and › signs as brackets
• Inclusion (set theory)
• Inequation
• Interval (mathematics)
• List of inequalities
• List of triangle inequalities
• Partially ordered set
• Relational operators, used in programming languages to denote inequality

References

1. "The Definitive Glossary of Higher Mathematical Jargon — Inequality". Math Vault. 2019-08-01. Retrieved 2019-12-03.
2. "Inequality Definition (Illustrated Mathematics Dictionary)". www.mathsisfun.com. Retrieved 2019-12-03.
3. "Inequality". www.learnalberta.ca. Retrieved 2019-12-03.
4. "Absolutely continuous measures - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-03.
5. Weisstein, Eric W. "Much Greater". mathworld.wolfram.com. Retrieved 2019-12-03.
6. Drachman, Bryon C.; Cloud, Michael J. (2006). Inequalities: With Applications to Engineering. Springer Science & Business Media. pp. 2–3. ISBN 0-3872-2626-5.
7. "ProvingInequalities". www.cs.yale.edu. Retrieved 2019-12-03.
8. Simovici, Dan A. & Djeraba, Chabane (2008). "Partially Ordered Sets". Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics. Springer. ISBN 9781848002012.
9. Weisstein, Eric W. "Partially Ordered Set". mathworld.wolfram.com. Retrieved 2019-12-03.
10. Feldman, Joel (2014). "Fields" (PDF). math.ubc.ca. Retrieved 2019-12-03.
11. Stewart, Ian (2007). Why Beauty Is Truth: The History of Symmetry. Hachette UK. p. 106. ISBN 978-0-4650-0875-9.
12. Brian W. Kernighan and Dennis M. Ritchie (Apr 1988). The C Programming Language. Prentice Hall Software Series (2nd ed.). Englewood Cliffs/NJ: Prentice Hall. ISBN 0131103628. Here: Sect.A.7.9 Relational Operators, p.167: Quote: "a<b<c is parsed as (a<b)<c"
13. Laub, M.; Ilani, Ishai (1990). "E3116". The American Mathematical Monthly. 97 (1): 65–67. doi:10.2307/2324012. JSTOR 2324012.
14. Manyama, S. (2010). "Solution of One Conjecture on Inequalities with Power-Exponential Functions" (PDF). Australian Journal of Mathematical Analysis and Applications. 7 (2): 1.
15. Gärtner, Bernd; Matoušek, Jiří (2006). Understanding and Using Linear Programming. Berlin: Springer. ISBN 3-540-30697-8.

Sources

• Hardy, G., Littlewood J. E., Pólya, G. (1999). Inequalities. Cambridge Mathematical Library, Cambridge University Press. ISBN 0-521-05206-8.
• Beckenbach, E. F., Bellman, R. (1975). An Introduction to Inequalities. Random House Inc. ISBN 0-394-01559-2.
• Drachman, Byron C., Cloud, Michael J. (1998). Inequalities: With Applications to Engineering. Springer-Verlag. ISBN 0-387-98404-6.
• Grinshpan, A. Z. (2005), "General inequalities, consequences, and applications", Advances in Applied Mathematics, 34 (1): 71–100, doi:10.1016/j.aam.2004.05.001
• Murray S. Klamkin. "'Quickie' inequalities" (PDF). Math Strategies.
• Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format.
• Harold Shapiro (2005). "Mathematical Problem Solving". The Old Problem Seminar. Kungliga Tekniska högskolan.
• "3rd USAMO". Archived from the original on 2008-02-03.
• Pachpatte, B. G. (2005). Mathematical Inequalities. North-Holland Mathematical Library. 67 (first ed.). Amsterdam, The Netherlands: Elsevier. ISBN 0-444-51795-2. ISSN 0924-6509. MR 2147066. Zbl 1091.26008.
• Ehrgott, Matthias (2005). Multicriteria Optimization. Springer-Berlin. ISBN 3-540-21398-8.
• Steele, J. Michael (2004). The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press. ISBN 978-0-521-54677-5.

External links

Wikimedia Commons has media related to Inequalities (mathematics).

• "Inequality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Graph of Inequalities by Ed Pegg, Jr.
• AoPS Wiki entry about Inequalities